For 2 positive integers m and n, such that m - n = 40, then which of the following is largest?
1. Square root (m+10) - Square root (n+10)
2. Square root (m+15) - Square root (n+15)
3. Square root (m+20) - Square root (n+20)
4. Square root (m+25) - Square root (n+25)
m – n = 40 or m = n + 40
Therefore,
Option 1: Square root (n+50) - Square root (n+10)
Option 2: Square root (n+55) - Square root (n+15)
Option 3: Square root (n+60) - Square root (n+20)
Option 4: Square root (n+65) - Square root (n+25)
Option 1:
Square root (n+50) - Square root (n+10)
= [(Square root (n+50) - Square root (n+10)) / (Square root (n+50) + Square root (n+10))] x [Square root (n+50) + Square root (n+10)]
= [40 / (Square root (n+50) + Square root (n+10))]
Similarly,
Option 2: = [40 / (Square root (n+55) + Square root (n+15))]
Option 3: = [40 / (Square root (n+60) - Square root (n+20))]
Option 4: = [40 / (Square root (n+65) - Square root (n+25))]
For all positive values of n, the denominator of option [1] is smallest and therefore option [1] is largest.